How do I copy existing properties in the UI so I can include them in my custom panel?
If you want the properties to be identical to the original ones, you should set Slider to False or simply not include Slider at all.

A Slider is a bar that fills the field from left to right. When disabled, the field has 2 arrows on both sides instead.

If you use a slider you have to include min and max values or else they will be determined by Blender and be huge.

Edit: Code would look like:

def draw(self, context):
    layout = self.layout

    cam = context.scene.camera

    col = layout.column(align=True)
    col.label(text="Location:")
    col.prop(cam, "location", index=0, text="X")
    col.prop(cam, "location", index=1, text="Y")
    col.prop(cam, "location", index=2, text="Z")
    col = layout.column(align=True)
    col.label(text="Rotation:")
    col.prop(cam, "rotation_euler", index=0, text="X")
    col.prop(cam, "rotation_euler", index=1, text="Y")
    col.prop(cam, "rotation_euler", index=2, text="Z")